LMFDB - Embedded newform 684.2.i.d.229.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [684,2,Mod(229,684)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(684, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 2, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("684.229");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 684 = 2^{2} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	684.i (of order $3$, degree $2$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$5.46176749826$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 18x^{14} + 123x^{12} + 399x^{10} + 631x^{8} + 465x^{6} + 153x^{4} + 21x^{2} + 1 $ x^16 + 18x^14 + 123x^12 + 399x^10 + 631x^8 + 465x^6 + 153x^4 + 21*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 3^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			229.5
Root			$-0.351047i$ of defining polynomial
Character	$\chi$	$=$	684.229
Dual form			684.2.i.d.457.5

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.276297 + 1.70987i) q^{3} +(0.304016 - 0.526570i) q^{5} +(-0.481049 - 0.833201i) q^{7} +(-2.84732 - 0.944864i) q^{9} +O(q^{10})$	\(q+(-0.276297 + 1.70987i) q^{3} +(0.304016 - 0.526570i) q^{5} +(-0.481049 - 0.833201i) q^{7} +(-2.84732 - 0.944864i) q^{9} +(1.96542 + 3.40420i) q^{11} +(-2.39243 + 4.14380i) q^{13} +(0.816369 + 0.665317i) q^{15} -6.28802 q^{17} +1.00000 q^{19} +(1.55758 - 0.592321i) q^{21} +(-0.996169 + 1.72542i) q^{23} +(2.31515 + 4.00996i) q^{25} +(2.40230 - 4.60749i) q^{27} +(2.05282 + 3.55558i) q^{29} +(-2.48593 + 4.30576i) q^{31} +(-6.36378 + 2.42004i) q^{33} -0.584985 q^{35} +4.71508 q^{37} +(-6.42435 - 5.23566i) q^{39} +(-5.84815 + 10.1293i) q^{41} +(-5.55976 - 9.62979i) q^{43} +(-1.36317 + 1.21206i) q^{45} +(-0.687309 - 1.19045i) q^{47} +(3.03718 - 5.26056i) q^{49} +(1.73736 - 10.7517i) q^{51} -13.4572 q^{53} +2.39007 q^{55} +(-0.276297 + 1.70987i) q^{57} +(0.655547 - 1.13544i) q^{59} +(2.09760 + 3.63315i) q^{61} +(0.582439 + 2.82692i) q^{63} +(1.45467 + 2.51956i) q^{65} +(6.00798 - 10.4061i) q^{67} +(-2.67500 - 2.18005i) q^{69} -8.54862 q^{71} -1.03696 q^{73} +(-7.49618 + 2.85067i) q^{75} +(1.89092 - 3.27517i) q^{77} +(6.25364 + 10.8316i) q^{79} +(7.21447 + 5.38066i) q^{81} +(5.77329 + 9.99963i) q^{83} +(-1.91166 + 3.31109i) q^{85} +(-6.64678 + 2.52766i) q^{87} +9.43217 q^{89} +4.60350 q^{91} +(-6.67544 - 5.44029i) q^{93} +(0.304016 - 0.526570i) q^{95} +(5.34045 + 9.24994i) q^{97} +(-2.37966 - 11.5499i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - q^{3} + 6 q^{5} - 5 q^{7} + 15 q^{9}+O(q^{10}) $ 16 * q - q^3 + 6 * q^5 - 5 * q^7 + 15 * q^9	$ 16 q - q^{3} + 6 q^{5} - 5 q^{7} + 15 q^{9} + 13 q^{11} - 3 q^{13} + 6 q^{15} - 20 q^{17} + 16 q^{19} + 3 q^{21} + 16 q^{23} - 14 q^{25} + 20 q^{27} + 3 q^{29} - 6 q^{31} - 17 q^{33} + 16 q^{37} - 16 q^{39} + 7 q^{41} - 3 q^{43} + 21 q^{45} + 10 q^{47} + 3 q^{49} - 2 q^{51} + 2 q^{53} - 12 q^{55} - q^{57} + 28 q^{59} - 7 q^{61} + 49 q^{63} + 21 q^{65} - 6 q^{67} - 56 q^{69} - 40 q^{71} - 20 q^{73} - 7 q^{75} + 22 q^{77} + 7 q^{79} + 39 q^{81} + 17 q^{83} + 24 q^{85} - 27 q^{87} - 32 q^{89} - 24 q^{91} - 22 q^{93} + 6 q^{95} + 11 q^{97} - q^{99}+O(q^{100}) $ 16 * q - q^3 + 6 * q^5 - 5 * q^7 + 15 * q^9 + 13 * q^11 - 3 * q^13 + 6 * q^15 - 20 * q^17 + 16 * q^19 + 3 * q^21 + 16 * q^23 - 14 * q^25 + 20 * q^27 + 3 * q^29 - 6 * q^31 - 17 * q^33 + 16 * q^37 - 16 * q^39 + 7 * q^41 - 3 * q^43 + 21 * q^45 + 10 * q^47 + 3 * q^49 - 2 * q^51 + 2 * q^53 - 12 * q^55 - q^57 + 28 * q^59 - 7 * q^61 + 49 * q^63 + 21 * q^65 - 6 * q^67 - 56 * q^69 - 40 * q^71 - 20 * q^73 - 7 * q^75 + 22 * q^77 + 7 * q^79 + 39 * q^81 + 17 * q^83 + 24 * q^85 - 27 * q^87 - 32 * q^89 - 24 * q^91 - 22 * q^93 + 6 * q^95 + 11 * q^97 - q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/684\mathbb{Z}\right)^\times$.

$n$	$325$	$343$	$533$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{3}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	−0.276297	+	1.70987i	−0.159520	+	0.987195i
$3$	−0.276297	+	1.70987i	−0.159520	+	0.987195i
$4$		0			0
$5$	0.304016	−	0.526570i	0.135960	−	0.235489i	−0.790004	−	0.613102i	$-0.789922\pi$
$5$	0.304016	−	0.526570i	0.135960	−	0.235489i	0.925964	+	0.377612i	$0.123255\pi$
$6$		0			0
$7$	−0.481049	−	0.833201i	−0.181819	−	0.314920i	0.760681	−	0.649126i	$-0.224865\pi$
$7$	−0.481049	−	0.833201i	−0.181819	−	0.314920i	−0.942500	+	0.334206i	$0.891532\pi$
$8$		0			0
$9$	−2.84732	−	0.944864i	−0.949107	−	0.314955i
$10$		0			0
$11$	1.96542	+	3.40420i	0.592595	+	1.02640i	0.993881	+	0.110452i	$0.0352298\pi$
$11$	1.96542	+	3.40420i	0.592595	+	1.02640i	−0.401287	+	0.915953i	$0.631437\pi$
$12$		0			0
$13$	−2.39243	+	4.14380i	−0.663540	+	1.14928i	0.316139	+	0.948713i	$0.397613\pi$
$13$	−2.39243	+	4.14380i	−0.663540	+	1.14928i	−0.979679	+	0.200572i	$0.935720\pi$
$14$		0			0
$15$	0.816369	+	0.665317i	0.210786	+	0.171784i
$16$		0			0
$17$	−6.28802			−1.52507			−0.762535	−	0.646947i	$-0.776046\pi$
$17$	−6.28802			−1.52507			−0.762535	+	0.646947i	$0.776046\pi$
$18$		0			0
$19$	1.00000			0.229416
$19$	1.00000			0.229416
$20$		0			0
$21$	1.55758	−	0.592321i	0.339892	−	0.129255i
$22$		0			0
$23$	−0.996169	+	1.72542i	−0.207716	+	0.359774i	−0.950995	−	0.309208i	$-0.899936\pi$
$23$	−0.996169	+	1.72542i	−0.207716	+	0.359774i	0.743279	+	0.668982i	$0.233270\pi$
$24$		0			0
$25$	2.31515	+	4.00996i	0.463030	+	0.801991i
$26$		0			0
$27$	2.40230	−	4.60749i	0.462323	−	0.886712i
$28$		0			0
$29$	2.05282	+	3.55558i	0.381199	+	0.660255i	0.991234	−	0.132119i	$-0.0421781\pi$
$29$	2.05282	+	3.55558i	0.381199	+	0.660255i	−0.610035	+	0.792374i	$0.708845\pi$
$30$		0			0
$31$	−2.48593	+	4.30576i	−0.446486	+	0.773337i	−0.998154	−	0.0607269i	$-0.980658\pi$
$31$	−2.48593	+	4.30576i	−0.446486	+	0.773337i	0.551668	+	0.834064i	$0.313991\pi$
$32$		0			0
$33$	−6.36378	+	2.42004i	−1.10779	+	0.421275i
$34$		0			0
$35$	−0.584985			−0.0988806
$36$		0			0
$37$	4.71508			0.775154			0.387577	−	0.921837i	$-0.373312\pi$
$37$	4.71508			0.775154			0.387577	+	0.921837i	$0.373312\pi$
$38$		0			0
$39$	−6.42435	−	5.23566i	−1.02872	−	0.838377i
$40$		0			0
$41$	−5.84815	+	10.1293i	−0.913327	+	1.58193i	−0.103995	+	0.994578i	$0.533162\pi$
$41$	−5.84815	+	10.1293i	−0.913327	+	1.58193i	−0.809332	+	0.587351i	$0.800171\pi$
$42$		0			0
$43$	−5.55976	−	9.62979i	−0.847856	−	1.46853i	−0.883118	−	0.469152i	$-0.844560\pi$
$43$	−5.55976	−	9.62979i	−0.847856	−	1.46853i	0.0352615	−	0.999378i	$-0.488774\pi$
$44$		0			0
$45$	−1.36317	+	1.21206i	−0.203209	+	0.180683i
$46$		0			0
$47$	−0.687309	−	1.19045i	−0.100254	−	0.173646i	0.811535	−	0.584304i	$-0.198632\pi$
$47$	−0.687309	−	1.19045i	−0.100254	−	0.173646i	−0.911789	+	0.410658i	$0.865299\pi$
$48$		0			0
$49$	3.03718	−	5.26056i	0.433883	−	0.751508i
$50$		0			0
$51$	1.73736	−	10.7517i	0.243279	−	1.50554i
$52$		0			0
$53$	−13.4572			−1.84849			−0.924243	−	0.381804i	$-0.875303\pi$
$53$	−13.4572			−1.84849			−0.924243	+	0.381804i	$0.875303\pi$
$54$		0			0
$55$	2.39007			0.322277
$56$		0			0
$57$	−0.276297	+	1.70987i	−0.0365964	+	0.226478i
$58$		0			0
$59$	0.655547	−	1.13544i	0.0853450	−	0.147822i	−0.820193	−	0.572087i	$-0.806134\pi$
$59$	0.655547	−	1.13544i	0.0853450	−	0.147822i	0.905538	+	0.424265i	$0.139467\pi$
$60$		0			0
$61$	2.09760	+	3.63315i	0.268571	+	0.465178i	0.968493	−	0.249041i	$-0.0801154\pi$
$61$	2.09760	+	3.63315i	0.268571	+	0.465178i	−0.699922	+	0.714219i	$0.746782\pi$
$62$		0			0
$63$	0.582439	+	2.82692i	0.0733804	+	0.356158i
$64$		0			0
$65$	1.45467	+	2.51956i	0.180430	+	0.312513i
$66$		0			0
$67$	6.00798	−	10.4061i	0.733992	−	1.27131i	−0.221172	−	0.975235i	$-0.570988\pi$
$67$	6.00798	−	10.4061i	0.733992	−	1.27131i	0.955164	−	0.296076i	$-0.0956783\pi$
$68$		0			0
$69$	−2.67500	−	2.18005i	−0.322032	−	0.262447i
$70$		0			0
$71$	−8.54862			−1.01453			−0.507267	−	0.861789i	$-0.669344\pi$
$71$	−8.54862			−1.01453			−0.507267	+	0.861789i	$0.669344\pi$
$72$		0			0
$73$	−1.03696			−0.121367			−0.0606837	−	0.998157i	$-0.519328\pi$
$73$	−1.03696			−0.121367			−0.0606837	+	0.998157i	$0.519328\pi$
$74$		0			0
$75$	−7.49618	+	2.85067i	−0.865584	+	0.329167i
$76$		0			0
$77$	1.89092	−	3.27517i	0.215491	−	0.373241i
$78$		0			0
$79$	6.25364	+	10.8316i	0.703590	+	1.21865i	0.967198	+	0.254024i	$0.0817541\pi$
$79$	6.25364	+	10.8316i	0.703590	+	1.21865i	−0.263608	+	0.964630i	$0.584913\pi$
$80$		0			0
$81$	7.21447	+	5.38066i	0.801607	+	0.597851i
$82$		0			0
$83$	5.77329	+	9.99963i	0.633701	+	1.09760i	0.986789	+	0.162012i	$0.0517984\pi$
$83$	5.77329	+	9.99963i	0.633701	+	1.09760i	−0.353088	+	0.935590i	$0.614868\pi$
$84$		0			0
$85$	−1.91166	+	3.31109i	−0.207348	+	0.359138i
$86$		0			0
$87$	−6.64678	+	2.52766i	−0.712609	+	0.270993i
$88$		0			0
$89$	9.43217			0.999808			0.499904	−	0.866081i	$-0.333369\pi$
$89$	9.43217			0.999808			0.499904	+	0.866081i	$0.333369\pi$
$90$		0			0
$91$	4.60350			0.482578
$92$		0			0
$93$	−6.67544	−	5.44029i	−0.692211	−	0.564131i
$94$		0			0
$95$	0.304016	−	0.526570i	0.0311913	−	0.0540250i
$96$		0			0
$97$	5.34045	+	9.24994i	0.542241	+	0.939189i	0.998775	+	0.0494829i	$0.0157573\pi$
$97$	5.34045	+	9.24994i	0.542241	+	0.939189i	−0.456534	+	0.889706i	$0.650909\pi$
$98$		0			0
$99$	−2.37966	−	11.5499i	−0.239165	−	1.16081i
$100$		0			0
$101$	1.37563	+	2.38266i	0.136880	+	0.237083i	0.926314	−	0.376752i	$-0.122959\pi$
$101$	1.37563	+	2.38266i	0.136880	+	0.237083i	−0.789434	+	0.613836i	$0.789626\pi$
$102$		0			0
$103$	8.46554	−	14.6627i	0.834135	−	1.44476i	−0.0605983	−	0.998162i	$-0.519301\pi$
$103$	8.46554	−	14.6627i	0.834135	−	1.44476i	0.894733	−	0.446601i	$-0.147366\pi$
$104$		0			0
$105$	0.161630	−	1.00025i	0.0157734	−	0.0976144i
$106$		0			0
$107$	10.2263			0.988614			0.494307	−	0.869287i	$-0.335422\pi$
$107$	10.2263			0.988614			0.494307	+	0.869287i	$0.335422\pi$
$108$		0			0
$109$	0.806983			0.0772950			0.0386475	−	0.999253i	$-0.487695\pi$
$109$	0.806983			0.0772950			0.0386475	+	0.999253i	$0.487695\pi$
$110$		0			0
$111$	−1.30276	+	8.06218i	−0.123653	+	0.765228i
$112$		0			0
$113$	1.27465	−	2.20776i	0.119909	−	0.207688i	−0.799823	−	0.600237i	$-0.795073\pi$
$113$	1.27465	−	2.20776i	0.119909	−	0.207688i	0.919731	+	0.392548i	$0.128406\pi$
$114$		0			0
$115$	0.605702	+	1.04911i	0.0564820	+	0.0978297i
$116$		0			0
$117$	10.7273	−	9.53822i	0.991742	−	0.881809i
$118$		0			0
$119$	3.02485	+	5.23919i	0.277287	+	0.480276i
$120$		0			0
$121$	−2.22571	+	3.85505i	−0.202338	+	0.350459i
$122$		0			0
$123$	−15.7040	−	12.7983i	−1.41598	−	1.15398i
$124$		0			0
$125$	5.85552			0.523734
$126$		0			0
$127$	21.4200			1.90072			0.950360	−	0.311151i	$-0.100715\pi$
$127$	21.4200			1.90072			0.950360	+	0.311151i	$0.100715\pi$
$128$		0			0
$129$	18.0019	−	6.84580i	1.58497	−	0.602739i
$130$		0			0
$131$	3.64506	−	6.31343i	0.318470	−	0.551607i	−0.661699	−	0.749770i	$-0.730164\pi$
$131$	3.64506	−	6.31343i	0.318470	−	0.551607i	0.980169	+	0.198163i	$0.0634975\pi$
$132$		0			0
$133$	−0.481049	−	0.833201i	−0.0417122	−	0.0722477i
$134$		0			0
$135$	−1.69583	−	2.66573i	−0.145954	−	0.229429i
$136$		0			0
$137$	−8.90516	−	15.4242i	−0.760819	−	1.31778i	−0.942429	−	0.334407i	$-0.891464\pi$
$137$	−8.90516	−	15.4242i	−0.760819	−	1.31778i	0.181610	−	0.983371i	$-0.441869\pi$
$138$		0			0
$139$	−0.299641	+	0.518993i	−0.0254152	+	0.0440204i	−0.878453	−	0.477828i	$-0.841424\pi$
$139$	−0.299641	+	0.518993i	−0.0254152	+	0.0440204i	0.853038	+	0.521849i	$0.174757\pi$
$140$		0			0
$141$	2.22543	−	0.846292i	0.187415	−	0.0712706i
$142$		0			0
$143$	−18.8084			−1.57284
$144$		0			0
$145$	2.49635			0.207311
$146$		0			0
$147$	8.15571	+	6.64667i	0.672672	+	0.548208i
$148$		0			0
$149$	−9.75491	+	16.8960i	−0.799153	+	1.38417i	0.121015	+	0.992651i	$0.461385\pi$
$149$	−9.75491	+	16.8960i	−0.799153	+	1.38417i	−0.920168	+	0.391524i	$0.871948\pi$
$150$		0			0
$151$	6.85015	+	11.8648i	0.557458	+	0.965545i	0.997708	+	0.0676696i	$0.0215564\pi$
$151$	6.85015	+	11.8648i	0.557458	+	0.965545i	−0.440250	+	0.897875i	$0.645110\pi$
$152$		0			0
$153$	17.9040	+	5.94133i	1.44745	+	0.480328i
$154$		0			0
$155$	1.51152	+	2.61803i	0.121408	+	0.210286i
$156$		0			0
$157$	0.465152	−	0.805666i	0.0371231	−	0.0642992i	−0.846867	−	0.531805i	$-0.821514\pi$
$157$	0.465152	−	0.805666i	0.0371231	−	0.0642992i	0.883990	+	0.467506i	$0.154847\pi$
$158$		0			0
$159$	3.71818	−	23.0101i	0.294871	−	1.82482i
$160$		0			0
$161$	1.91682			0.151067
$162$		0			0
$163$	10.9550			0.858059			0.429029	−	0.903291i	$-0.358856\pi$
$163$	10.9550			0.858059			0.429029	+	0.903291i	$0.358856\pi$
$164$		0			0
$165$	−0.660368	+	4.08671i	−0.0514096	+	0.318150i
$166$		0			0
$167$	0.688389	−	1.19232i	0.0532691	−	0.0922648i	−0.838161	−	0.545423i	$-0.816369\pi$
$167$	0.688389	−	1.19232i	0.0532691	−	0.0922648i	0.891430	+	0.453158i	$0.149703\pi$
$168$		0			0
$169$	−4.94741	−	8.56916i	−0.380570	−	0.659166i
$170$		0			0
$171$	−2.84732	−	0.944864i	−0.217740	−	0.0722555i
$172$		0			0
$173$	6.82768	+	11.8259i	0.519099	+	0.899106i	0.999754	+	0.0221958i	$0.00706571\pi$
$173$	6.82768	+	11.8259i	0.519099	+	0.899106i	−0.480655	+	0.876910i	$0.659601\pi$
$174$		0			0
$175$	2.22740	−	3.85797i	0.168376	−	0.291635i
$176$		0			0
$177$	1.76033	+	1.43462i	0.132315	+	0.107833i
$178$		0			0
$179$	−9.03997			−0.675679			−0.337840	−	0.941204i	$-0.609696\pi$
$179$	−9.03997			−0.675679			−0.337840	+	0.941204i	$0.609696\pi$
$180$		0			0
$181$	−2.41496			−0.179503			−0.0897513	−	0.995964i	$-0.528607\pi$
$181$	−2.41496			−0.179503			−0.0897513	+	0.995964i	$0.528607\pi$
$182$		0			0
$183$	−6.79179	+	2.58280i	−0.502063	+	0.190926i
$184$		0			0
$185$	1.43346	−	2.48282i	0.105390	−	0.182541i
$186$		0			0
$187$	−12.3586	−	21.4057i	−0.903749	−	1.56534i
$188$		0			0
$189$	−4.99459	+	0.214828i	−0.363303	+	0.0156264i
$190$		0			0
$191$	1.62965	+	2.82264i	0.117918	+	0.204239i	0.918942	−	0.394392i	$-0.129045\pi$
$191$	1.62965	+	2.82264i	0.117918	+	0.204239i	−0.801025	+	0.598631i	$0.795711\pi$
$192$		0			0
$193$	3.67589	−	6.36683i	0.264597	−	0.458295i	−0.702861	−	0.711327i	$-0.748095\pi$
$193$	3.67589	−	6.36683i	0.264597	−	0.458295i	0.967458	+	0.253032i	$0.0814279\pi$
$194$		0			0
$195$	−4.71005	+	1.79115i	−0.337293	+	0.128267i
$196$		0			0
$197$	−11.6201			−0.827900			−0.413950	−	0.910300i	$-0.635851\pi$
$197$	−11.6201			−0.827900			−0.413950	+	0.910300i	$0.635851\pi$
$198$		0			0
$199$	−25.2837			−1.79232			−0.896158	−	0.443735i	$-0.853653\pi$
$199$	−25.2837			−1.79232			−0.896158	+	0.443735i	$0.853653\pi$
$200$		0			0
$201$	16.1332	+	13.1481i	1.13795	+	0.927392i
$202$		0			0
$203$	1.97501	−	3.42082i	0.138619	−	0.240095i
$204$		0			0
$205$	3.55585	+	6.15892i	0.248352	+	0.430158i
$206$		0			0
$207$	4.46669	−	3.97157i	0.310457	−	0.276043i
$208$		0			0
$209$	1.96542	+	3.40420i	0.135951	+	0.235473i
$210$		0			0
$211$	8.61463	−	14.9210i	0.593056	−	1.02720i	−0.400762	−	0.916182i	$-0.631255\pi$
$211$	8.61463	−	14.9210i	0.593056	−	1.02720i	0.993818	−	0.111021i	$-0.0354120\pi$
$212$		0			0
$213$	2.36196	−	14.6170i	0.161839	−	1.00154i
$214$		0			0
$215$	−6.76102			−0.461098
$216$		0			0
$217$	4.78342			0.324719
$218$		0			0
$219$	0.286509	−	1.77307i	0.0193605	−	0.119813i
$220$		0			0
$221$	15.0436	−	26.0563i	1.01194	−	1.75274i
$222$		0			0
$223$	−4.21546	−	7.30139i	−0.282288	−	0.488938i	0.689660	−	0.724134i	$-0.257760\pi$
$223$	−4.21546	−	7.30139i	−0.282288	−	0.488938i	−0.971948	+	0.235196i	$0.924427\pi$
$224$		0			0
$225$	−2.80311	−	13.6051i	−0.186874	−	0.907009i
$226$		0			0
$227$	7.60868	+	13.1786i	0.505006	+	0.874695i	0.999983	+	0.00578967i	$0.00184292\pi$
$227$	7.60868	+	13.1786i	0.505006	+	0.874695i	−0.494978	+	0.868906i	$0.664824\pi$
$228$		0			0
$229$	−10.8847	+	18.8529i	−0.719281	+	1.24583i	0.242003	+	0.970275i	$0.422195\pi$
$229$	−10.8847	+	18.8529i	−0.719281	+	1.24583i	−0.961285	+	0.275557i	$0.911138\pi$
$230$		0			0
$231$	5.07767	+	4.13815i	0.334086	+	0.272270i
$232$		0			0
$233$	6.33257			0.414860			0.207430	−	0.978250i	$-0.433490\pi$
$233$	6.33257			0.414860			0.207430	+	0.978250i	$0.433490\pi$
$234$		0			0
$235$	−0.835811			−0.0545223
$236$		0			0
$237$	−20.2486	+	7.70018i	−1.31528	+	0.500181i
$238$		0			0
$239$	11.8893	−	20.5929i	0.769058	−	1.33205i	−0.169017	−	0.985613i	$-0.554059\pi$
$239$	11.8893	−	20.5929i	0.769058	−	1.33205i	0.938074	−	0.346434i	$-0.112608\pi$
$240$		0			0
$241$	0.686731	+	1.18945i	0.0442363	+	0.0766195i	0.887296	−	0.461201i	$-0.152581\pi$
$241$	0.686731	+	1.18945i	0.0442363	+	0.0766195i	−0.843060	+	0.537820i	$0.819248\pi$
$242$		0			0
$243$	−11.1936	+	10.8491i	−0.718068	+	0.695973i
$244$		0			0
$245$	−1.84670	−	3.19858i	−0.117981	−	0.204350i
$246$		0			0
$247$	−2.39243	+	4.14380i	−0.152226	+	0.263664i
$248$		0			0
$249$	−18.6932	+	7.10872i	−1.18464	+	0.450497i
$250$		0			0
$251$	−7.92155			−0.500004			−0.250002	−	0.968245i	$-0.580431\pi$
$251$	−7.92155			−0.500004			−0.250002	+	0.968245i	$0.580431\pi$
$252$		0			0
$253$	−7.83154			−0.492365
$254$		0			0
$255$	−5.13335	−	4.18353i	−0.321463	−	0.261983i
$256$		0			0
$257$	9.07426	−	15.7171i	0.566037	−	0.980405i	−0.430915	−	0.902392i	$-0.641809\pi$
$257$	9.07426	−	15.7171i	0.566037	−	0.980405i	0.996952	−	0.0780125i	$-0.0248574\pi$
$258$		0			0
$259$	−2.26818	−	3.92861i	−0.140938	−	0.244112i
$260$		0			0
$261$	−2.48549	−	12.0635i	−0.153848	−	0.746713i
$262$		0			0
$263$	9.58536	+	16.6023i	0.591059	+	1.02374i	0.994090	+	0.108558i	$0.0346232\pi$
$263$	9.58536	+	16.6023i	0.591059	+	1.02374i	−0.403031	+	0.915186i	$0.632043\pi$
$264$		0			0
$265$	−4.09119	+	7.08615i	−0.251320	+	0.435299i
$266$		0			0
$267$	−2.60608	+	16.1278i	−0.159489	+	0.987005i
$268$		0			0
$269$	−7.34276			−0.447696			−0.223848	−	0.974624i	$-0.571862\pi$
$269$	−7.34276			−0.447696			−0.223848	+	0.974624i	$0.571862\pi$
$270$		0			0
$271$	−5.88386			−0.357419			−0.178710	−	0.983902i	$-0.557192\pi$
$271$	−5.88386			−0.357419			−0.178710	+	0.983902i	$0.557192\pi$
$272$		0			0
$273$	−1.27193	+	7.87139i	−0.0769808	+	0.476398i
$274$		0			0
$275$	−9.10046	+	15.7625i	−0.548778	+	0.950512i
$276$		0			0
$277$	1.46659	+	2.54020i	0.0881186	+	0.152626i	0.906716	−	0.421742i	$-0.138581\pi$
$277$	1.46659	+	2.54020i	0.0881186	+	0.152626i	−0.818597	+	0.574368i	$0.805248\pi$
$278$		0			0
$279$	11.1466	−	9.91100i	0.667329	−	0.593356i
$280$		0			0
$281$	7.09516	+	12.2892i	0.423262	+	0.733111i	0.996256	−	0.0864484i	$-0.0275518\pi$
$281$	7.09516	+	12.2892i	0.423262	+	0.733111i	−0.572995	+	0.819559i	$0.694218\pi$
$282$		0			0
$283$	1.89945	−	3.28994i	0.112911	−	0.195567i	−0.804032	−	0.594586i	$-0.797316\pi$
$283$	1.89945	−	3.28994i	0.112911	−	0.195567i	0.916943	+	0.399019i	$0.130649\pi$
$284$		0			0
$285$	0.816369	+	0.665317i	0.0483575	+	0.0394100i
$286$		0			0
$287$	11.2530			0.664242
$288$		0			0
$289$	22.5393			1.32584
$290$		0			0
$291$	−17.2918	+	6.57576i	−1.01366	+	0.385478i
$292$		0			0
$293$	5.14981	−	8.91972i	0.300855	−	0.521096i	−0.675475	−	0.737383i	$-0.736061\pi$
$293$	5.14981	−	8.91972i	0.300855	−	0.521096i	0.976330	+	0.216287i	$0.0693947\pi$
$294$		0			0
$295$	−0.398593	−	0.690383i	−0.0232070	−	0.0401957i
$296$		0			0
$297$	20.4063	−	0.877719i	1.18410	−	0.0509304i
$298$		0			0
$299$	−4.76652	−	8.25586i	−0.275655	−	0.477449i
$300$		0			0
$301$	−5.34904	+	9.26480i	−0.308313	+	0.534014i
$302$		0			0
$303$	−4.45412	+	1.69383i	−0.255883	+	0.0973078i
$304$		0			0
$305$	2.55082			0.146059
$306$		0			0
$307$	−9.42854			−0.538115			−0.269058	−	0.963124i	$-0.586712\pi$
$307$	−9.42854			−0.538115			−0.269058	+	0.963124i	$0.586712\pi$
$308$		0			0
$309$	22.7324	+	18.5263i	1.29320	+	1.05392i
$310$		0			0
$311$	−10.1603	+	17.5982i	−0.576140	+	0.997904i	0.419777	+	0.907627i	$0.362108\pi$
$311$	−10.1603	+	17.5982i	−0.576140	+	0.997904i	−0.995917	+	0.0902762i	$0.971225\pi$
$312$		0			0
$313$	7.52479	+	13.0333i	0.425326	+	0.736686i	0.996451	−	0.0841769i	$-0.0268261\pi$
$313$	7.52479	+	13.0333i	0.425326	+	0.736686i	−0.571125	+	0.820863i	$0.693493\pi$
$314$		0			0
$315$	1.66564	+	0.552732i	0.0938482	+	0.0311429i
$316$		0			0
$317$	−4.83178	−	8.36889i	−0.271380	−	0.470044i	0.697836	−	0.716258i	$-0.254147\pi$
$317$	−4.83178	−	8.36889i	−0.271380	−	0.470044i	−0.969215	+	0.246214i	$0.920813\pi$
$318$		0			0
$319$	−8.06928	+	13.9764i	−0.451793	+	0.782528i
$320$		0			0
$321$	−2.82549	+	17.4857i	−0.157704	+	0.975955i
$322$		0			0
$323$	−6.28802			−0.349875
$324$		0			0
$325$	−22.1553			−1.22895
$326$		0			0
$327$	−0.222967	+	1.37984i	−0.0123301	+	0.0763052i
$328$		0			0
$329$	−0.661259	+	1.14533i	−0.0364564	+	0.0631443i
$330$		0			0
$331$	1.12700	+	1.95202i	0.0619454	+	0.107293i	0.895335	−	0.445393i	$-0.146936\pi$
$331$	1.12700	+	1.95202i	0.0619454	+	0.107293i	−0.833390	+	0.552686i	$0.813603\pi$
$332$		0			0
$333$	−13.4253	−	4.45511i	−0.735704	−	0.244138i
$334$		0			0
$335$	−3.65304	−	6.32725i	−0.199587	−	0.345695i
$336$		0			0
$337$	−17.9645	+	31.1155i	−0.978590	+	1.69497i	−0.311052	+	0.950393i	$0.600681\pi$
$337$	−17.9645	+	31.1155i	−0.978590	+	1.69497i	−0.667539	+	0.744575i	$0.732652\pi$
$338$		0			0
$339$	3.42280	+	2.78948i	0.185901	+	0.151504i
$340$		0			0
$341$	−19.5435			−1.05834
$342$		0			0
$343$	−12.5788			−0.679193
$344$		0			0
$345$	−1.96119	+	0.745808i	−0.105587	+	0.0401529i
$346$		0			0
$347$	14.9321	−	25.8631i	0.801596	−	1.38841i	−0.116969	−	0.993136i	$-0.537318\pi$
$347$	14.9321	−	25.8631i	0.801596	−	1.38841i	0.918565	−	0.395270i	$-0.129349\pi$
$348$		0			0
$349$	−5.46531	−	9.46619i	−0.292551	−	0.506713i	0.681861	−	0.731482i	$-0.261171\pi$
$349$	−5.46531	−	9.46619i	−0.292551	−	0.506713i	−0.974412	+	0.224768i	$0.927838\pi$
$350$		0			0
$351$	13.3452	+	20.9777i	0.712314	+	1.11971i
$352$		0			0
$353$	1.84814	+	3.20108i	0.0983667	+	0.170376i	0.911009	−	0.412387i	$-0.135305\pi$
$353$	1.84814	+	3.20108i	0.0983667	+	0.170376i	−0.812642	+	0.582763i	$0.801972\pi$
$354$		0			0
$355$	−2.59891	+	4.50145i	−0.137936	+	0.238912i
$356$		0			0
$357$	−9.79410	+	3.72453i	−0.518359	+	0.197123i
$358$		0			0
$359$	−0.951053			−0.0501946			−0.0250973	−	0.999685i	$-0.507990\pi$
$359$	−0.951053			−0.0501946			−0.0250973	+	0.999685i	$0.507990\pi$
$360$		0			0
$361$	1.00000			0.0526316
$362$		0			0
$363$	−5.97668	−	4.87082i	−0.313694	−	0.255652i
$364$		0			0
$365$	−0.315253	+	0.546034i	−0.0165011	+	0.0285807i
$366$		0			0
$367$	1.86291	+	3.22666i	0.0972433	+	0.168430i	0.910543	−	0.413415i	$-0.135664\pi$
$367$	1.86291	+	3.22666i	0.0972433	+	0.168430i	−0.813299	+	0.581846i	$0.802331\pi$
$368$		0			0
$369$	26.2223	−	23.3156i	1.36508	−	1.21376i
$370$		0			0
$371$	6.47356	+	11.2125i	0.336091	+	0.582126i
$372$		0			0
$373$	15.4631	−	26.7829i	0.800649	−	1.38676i	−0.118541	−	0.992949i	$-0.537822\pi$
$373$	15.4631	−	26.7829i	0.800649	−	1.38676i	0.919190	−	0.393815i	$-0.128845\pi$
$374$		0			0
$375$	−1.61786	+	10.0122i	−0.0835460	+	0.517027i
$376$		0			0
$377$	−19.6449			−1.01176
$378$		0			0
$379$	25.8416			1.32739			0.663696	−	0.748002i	$-0.268987\pi$
$379$	25.8416			1.32739			0.663696	+	0.748002i	$0.268987\pi$
$380$		0			0
$381$	−5.91828	+	36.6255i	−0.303203	+	1.87638i
$382$		0			0
$383$	2.57799	−	4.46522i	0.131729	−	0.228162i	−0.792614	−	0.609724i	$-0.791280\pi$
$383$	2.57799	−	4.46522i	0.131729	−	0.228162i	0.924343	+	0.381562i	$0.124614\pi$
$384$		0			0
$385$	−1.14974	−	1.99141i	−0.0585961	−	0.101491i
$386$		0			0
$387$	6.73159	+	32.6723i	0.342186	+	1.66083i
$388$		0			0
$389$	−13.4868	−	23.3598i	−0.683807	−	1.18439i	−0.973810	−	0.227362i	$-0.926990\pi$
$389$	−13.4868	−	23.3598i	−0.683807	−	1.18439i	0.290003	−	0.957026i	$-0.406344\pi$
$390$		0			0
$391$	6.26394	−	10.8495i	0.316781	−	0.548680i
$392$		0			0
$393$	9.78803	+	7.97696i	0.493741	+	0.402385i
$394$		0			0
$395$	7.60482			0.382640
$396$		0			0
$397$	−37.8661			−1.90045			−0.950224	−	0.311569i	$-0.899146\pi$
$397$	−37.8661			−1.90045			−0.950224	+	0.311569i	$0.899146\pi$
$398$		0			0
$399$	1.55758	−	0.592321i	0.0779765	−	0.0296531i
$400$		0			0
$401$	−5.74431	+	9.94943i	−0.286857	+	0.496851i	−0.973058	−	0.230561i	$-0.925944\pi$
$401$	−5.74431	+	9.94943i	−0.286857	+	0.496851i	0.686201	+	0.727412i	$0.259277\pi$
$402$		0			0
$403$	−11.8948	−	20.6024i	−0.592523	−	1.02628i
$404$		0			0
$405$	5.02661	−	2.16312i	0.249774	−	0.107486i
$406$		0			0
$407$	9.26709	+	16.0511i	0.459352	+	0.795622i
$408$		0			0
$409$	−3.69092	+	6.39286i	−0.182504	+	0.316106i	−0.942733	−	0.333549i	$-0.891754\pi$
$409$	−3.69092	+	6.39286i	−0.182504	+	0.316106i	0.760229	+	0.649656i	$0.225087\pi$
$410$		0			0
$411$	28.8338	−	10.9650i	1.42227	−	0.540865i
$412$		0			0
$413$	−1.26140			−0.0620695
$414$		0			0
$415$	7.02068			0.344632
$416$		0			0
$417$	−0.804622	−	0.655744i	−0.0394025	−	0.0321119i
$418$		0			0
$419$	−6.18622	+	10.7149i	−0.302217	+	0.523455i	−0.976638	−	0.214892i	$-0.931060\pi$
$419$	−6.18622	+	10.7149i	−0.302217	+	0.523455i	0.674421	+	0.738347i	$0.264393\pi$
$420$		0			0
$421$	5.22231	+	9.04531i	0.254520	+	0.440842i	0.964765	−	0.263113i	$-0.0847492\pi$
$421$	5.22231	+	9.04531i	0.254520	+	0.440842i	−0.710245	+	0.703955i	$0.751416\pi$
$422$		0			0
$423$	0.832172	+	4.03902i	0.0404616	+	0.196384i
$424$		0			0
$425$	−14.5577	−	25.2147i	−0.706153	−	1.22309i
$426$		0			0
$427$	2.01810	−	3.49545i	0.0976627	−	0.169157i
$428$		0			0
$429$	5.19671	−	32.1600i	0.250900	−	1.55270i
$430$		0			0
$431$	27.8693			1.34242			0.671208	−	0.741269i	$-0.265776\pi$
$431$	27.8693			1.34242			0.671208	+	0.741269i	$0.265776\pi$
$432$		0			0
$433$	−38.7466			−1.86204			−0.931021	−	0.364966i	$-0.881081\pi$
$433$	−38.7466			−1.86204			−0.931021	+	0.364966i	$0.881081\pi$
$434$		0			0
$435$	−0.689734	+	4.26844i	−0.0330702	+	0.204656i
$436$		0			0
$437$	−0.996169	+	1.72542i	−0.0476532	+	0.0825378i
$438$		0			0
$439$	13.7979	+	23.8986i	0.658536	+	1.14062i	0.980995	+	0.194034i	$0.0621572\pi$
$439$	13.7979	+	23.8986i	0.658536	+	1.14062i	−0.322459	+	0.946583i	$0.604509\pi$
$440$		0			0
$441$	−13.6183	+	12.1088i	−0.648493	+	0.576608i
$442$		0			0
$443$	0.978766	+	1.69527i	0.0465026	+	0.0805448i	0.888340	−	0.459187i	$-0.151859\pi$
$443$	0.978766	+	1.69527i	0.0465026	+	0.0805448i	−0.841837	+	0.539731i	$0.818526\pi$
$444$		0			0
$445$	2.86753	−	4.96670i	0.135934	−	0.235444i
$446$		0			0
$447$	−26.1947	−	21.3480i	−1.23897	−	1.00972i
$448$		0			0
$449$	−3.62439			−0.171046			−0.0855228	−	0.996336i	$-0.527256\pi$
$449$	−3.62439			−0.171046			−0.0855228	+	0.996336i	$0.527256\pi$
$450$		0			0
$451$	−45.9761			−2.16493
$452$		0			0
$453$	−22.1800	+	8.43467i	−1.04211	+	0.396295i
$454$		0			0
$455$	1.39953	−	2.42407i	0.0656112	−	0.113642i
$456$		0			0
$457$	−15.0709	−	26.1035i	−0.704986	−	1.22107i	−0.966697	−	0.255925i	$-0.917620\pi$
$457$	−15.0709	−	26.1035i	−0.704986	−	1.22107i	0.261710	−	0.965146i	$-0.415714\pi$
$458$		0			0
$459$	−15.1057	+	28.9720i	−0.705075	+	1.35230i
$460$		0			0
$461$	−3.60557	−	6.24503i	−0.167928	−	0.290860i	0.769763	−	0.638330i	$-0.220374\pi$
$461$	−3.60557	−	6.24503i	−0.167928	−	0.290860i	−0.937691	+	0.347470i	$0.887041\pi$
$462$		0			0
$463$	−2.94688	+	5.10415i	−0.136953	+	0.237210i	−0.926342	−	0.376684i	$-0.877064\pi$
$463$	−2.94688	+	5.10415i	−0.136953	+	0.237210i	0.789389	+	0.613894i	$0.210398\pi$
$464$		0			0
$465$	−4.89413	+	1.86116i	−0.226960	+	0.0863090i
$466$		0			0
$467$	14.3747			0.665182			0.332591	−	0.943071i	$-0.392077\pi$
$467$	14.3747			0.665182			0.332591	+	0.943071i	$0.392077\pi$
$468$		0			0
$469$	−11.5605			−0.533816
$470$		0			0
$471$	1.24907	+	1.01795i	0.0575539	+	0.0469048i
$472$		0			0
$473$	21.8545	−	37.8531i	1.00487	−	1.74049i
$474$		0			0
$475$	2.31515	+	4.00996i	0.106226	+	0.183989i
$476$		0			0
$477$	38.3169	+	12.7152i	1.75441	+	0.582189i
$478$		0			0
$479$	16.2622	+	28.1669i	0.743037	+	1.28698i	0.951106	+	0.308864i	$0.0999488\pi$
$479$	16.2622	+	28.1669i	0.743037	+	1.28698i	−0.208069	+	0.978114i	$0.566718\pi$
$480$		0			0
$481$	−11.2805	+	19.5384i	−0.514346	+	0.890873i
$482$		0			0
$483$	−0.529612	+	3.27752i	−0.0240982	+	0.149132i
$484$		0			0
$485$	6.49432			0.294892
$486$		0			0
$487$	−27.9399			−1.26608			−0.633038	−	0.774121i	$-0.718192\pi$
$487$	−27.9399			−1.26608			−0.633038	+	0.774121i	$0.718192\pi$
$488$		0			0
$489$	−3.02682	+	18.7316i	−0.136878	+	0.847071i
$490$		0			0
$491$	12.8143	−	22.1950i	0.578300	−	1.00164i	−0.417375	−	0.908734i	$-0.637050\pi$
$491$	12.8143	−	22.1950i	0.578300	−	1.00164i	0.995675	−	0.0929098i	$-0.0296168\pi$
$492$		0			0
$493$	−12.9082	−	22.3576i	−0.581355	−	1.00694i
$494$		0			0
$495$	−6.80529	−	2.25829i	−0.305875	−	0.101502i
$496$		0			0
$497$	4.11231	+	7.12272i	0.184462	+	0.319498i
$498$		0			0
$499$	−8.81725	+	15.2719i	−0.394714	+	0.683665i	−0.993065	−	0.117569i	$-0.962490\pi$
$499$	−8.81725	+	15.2719i	−0.394714	+	0.683665i	0.598350	+	0.801234i	$0.295823\pi$
$500$		0			0
$501$	1.84852	+	1.50649i	0.0825859	+	0.0673051i
$502$		0			0
$503$	−42.8959			−1.91264			−0.956318	−	0.292330i	$-0.905570\pi$
$503$	−42.8959			−1.91264			−0.956318	+	0.292330i	$0.905570\pi$
$504$		0			0
$505$	1.67285			0.0744409
$506$		0			0
$507$	16.0191	−	6.09180i	0.711434	−	0.270546i
$508$		0			0
$509$	12.5852	−	21.7983i	0.557831	−	0.966192i	−0.439846	−	0.898073i	$-0.644967\pi$
$509$	12.5852	−	21.7983i	0.557831	−	0.966192i	0.997677	−	0.0681189i	$-0.0216997\pi$
$510$		0			0
$511$	0.498830	+	0.863999i	0.0220669	+	0.0382211i
$512$		0			0
$513$	2.40230	−	4.60749i	0.106064	−	0.203426i
$514$		0			0
$515$	−5.14731	−	8.91541i	−0.226818	−	0.392860i
$516$		0			0
$517$	2.70170	−	4.67947i	0.118820	−	0.205803i
$518$		0			0
$519$	−22.1072	+	8.40700i	−0.970399	+	0.369026i
$520$		0			0
$521$	20.0612			0.878895			0.439448	−	0.898268i	$-0.355174\pi$
$521$	20.0612			0.878895			0.439448	+	0.898268i	$0.355174\pi$
$522$		0			0
$523$	23.4609			1.02588			0.512938	−	0.858426i	$-0.328557\pi$
$523$	23.4609			1.02588			0.512938	+	0.858426i	$0.328557\pi$
$524$		0			0
$525$	5.98121	+	4.87451i	0.261041	+	0.212741i
$526$		0			0
$527$	15.6316	−	27.0747i	0.680923	−	1.17939i
$528$		0			0
$529$	9.51529	+	16.4810i	0.413708	+	0.716564i
$530$		0			0
$531$	−2.93939	+	2.61356i	−0.127559	+	0.113419i
$532$		0			0
$533$	−27.9825	−	48.4671i	−1.21206	−	2.09934i
$534$		0			0
$535$	3.10896	−	5.38487i	0.134412	−	0.232808i
$536$		0			0
$537$	2.49771	−	15.4572i	0.107784	−	0.667027i
$538$		0			0
$539$	23.8773			1.02847
$540$		0			0
$541$	14.7989			0.636256			0.318128	−	0.948048i	$-0.396946\pi$
$541$	14.7989			0.636256			0.318128	+	0.948048i	$0.396946\pi$
$542$		0			0
$543$	0.667246	−	4.12927i	0.0286343	−	0.177204i
$544$		0			0
$545$	0.245336	−	0.424934i	0.0105090	−	0.0182022i
$546$		0			0
$547$	−2.00041	−	3.46481i	−0.0855312	−	0.148144i	0.820086	−	0.572240i	$-0.193925\pi$
$547$	−2.00041	−	3.46481i	−0.0855312	−	0.148144i	−0.905617	+	0.424096i	$0.860592\pi$
$548$		0			0
$549$	−2.53971	−	12.3267i	−0.108392	−	0.526091i
$550$		0			0
$551$	2.05282	+	3.55558i	0.0874530	+	0.151473i
$552$		0			0
$553$	6.01662	−	10.4211i	0.255853	−	0.443150i
$554$		0			0
$555$	3.84925	+	3.13702i	0.163391	+	0.133159i
$556$		0			0
$557$	−15.5868			−0.660433			−0.330217	−	0.943905i	$-0.607122\pi$
$557$	−15.5868			−0.660433			−0.330217	+	0.943905i	$0.607122\pi$
$558$		0			0
$559$	53.2053			2.25034
$560$		0			0
$561$	40.0156	−	15.2173i	1.68946	−	0.642473i
$562$		0			0
$563$	4.55037	−	7.88148i	0.191775	−	0.332165i	−0.754063	−	0.656802i	$-0.771909\pi$
$563$	4.55037	−	7.88148i	0.191775	−	0.332165i	0.945839	+	0.324637i	$0.105242\pi$
$564$		0			0
$565$	−0.775026	−	1.34238i	−0.0326056	−	0.0564745i
$566$		0			0
$567$	1.01266	−	8.59946i	0.0425277	−	0.361143i
$568$		0			0
$569$	5.68244	+	9.84227i	0.238220	+	0.412610i	0.960204	−	0.279301i	$-0.0901027\pi$
$569$	5.68244	+	9.84227i	0.238220	+	0.412610i	−0.721983	+	0.691910i	$0.756769\pi$
$570$		0			0
$571$	4.18985	−	7.25704i	0.175340	−	0.303698i	−0.764939	−	0.644103i	$-0.777231\pi$
$571$	4.18985	−	7.25704i	0.175340	−	0.303698i	0.940279	+	0.340405i	$0.110564\pi$
$572$		0			0
$573$	−5.27662	+	2.00661i	−0.220434	+	0.0838273i
$574$		0			0
$575$	−9.22512			−0.384714
$576$		0			0
$577$	21.7058			0.903622			0.451811	−	0.892114i	$-0.350778\pi$
$577$	21.7058			0.903622			0.451811	+	0.892114i	$0.350778\pi$
$578$		0			0
$579$	9.87083	+	8.04444i	0.410218	+	0.334315i
$580$		0			0
$581$	5.55447	−	9.62063i	0.230438	−	0.399131i
$582$		0			0
$583$	−26.4490	−	45.8109i	−1.09540	−	1.89729i
$584$		0			0
$585$	−1.76127	−	8.54846i	−0.0728195	−	0.353435i
$586$		0			0
$587$	16.2665	+	28.1744i	0.671390	+	1.16288i	0.977510	+	0.210889i	$0.0676359\pi$
$587$	16.2665	+	28.1744i	0.671390	+	1.16288i	−0.306120	+	0.951993i	$0.599031\pi$
$588$		0			0
$589$	−2.48593	+	4.30576i	−0.102431	+	0.177416i
$590$		0			0
$591$	3.21060	−	19.8689i	0.132067	−	0.817299i
$592$		0			0
$593$	33.4141			1.37215			0.686076	−	0.727530i	$-0.259332\pi$
$593$	33.4141			1.37215			0.686076	+	0.727530i	$0.259332\pi$
$594$		0			0
$595$	3.67840			0.150800
$596$		0			0
$597$	6.98581	−	43.2319i	0.285910	−	1.76937i
$598$		0			0
$599$	−10.9185	+	18.9114i	−0.446118	+	0.772699i	−0.998129	−	0.0611376i	$-0.980527\pi$
$599$	−10.9185	+	18.9114i	−0.446118	+	0.772699i	0.552011	+	0.833837i	$0.313860\pi$
$600$		0			0
$601$	4.89564	+	8.47949i	0.199697	+	0.345886i	0.948430	−	0.316986i	$-0.102671\pi$
$601$	4.89564	+	8.47949i	0.199697	+	0.345886i	−0.748733	+	0.662872i	$0.769337\pi$
$602$		0			0
$603$	−26.9390	+	23.9529i	−1.09704	+	0.975436i
$604$		0			0
$605$	1.35330	+	2.34399i	0.0550196	+	0.0952967i
$606$		0			0
$607$	−0.641483	+	1.11108i	−0.0260370	+	0.0450974i	−0.878750	−	0.477282i	$-0.841622\pi$
$607$	−0.641483	+	1.11108i	−0.0260370	+	0.0450974i	0.852713	+	0.522379i	$0.174955\pi$
$608$		0			0
$609$	5.30347	+	4.32218i	0.214908	+	0.175143i
$610$		0			0
$611$	6.57735			0.266091
$612$		0			0
$613$	−44.3227			−1.79018			−0.895088	−	0.445890i	$-0.852887\pi$
$613$	−44.3227			−1.79018			−0.895088	+	0.445890i	$0.852887\pi$
$614$		0			0
$615$	−11.5134	+	4.37836i	−0.464266	+	0.176553i
$616$		0			0
$617$	−21.4340	+	37.1248i	−0.862902	+	1.49459i	0.00621355	+	0.999981i	$0.498022\pi$
$617$	−21.4340	+	37.1248i	−0.862902	+	1.49459i	−0.869115	+	0.494609i	$0.835311\pi$
$618$		0			0
$619$	−6.73371	−	11.6631i	−0.270651	−	0.468781i	0.698378	−	0.715729i	$-0.253906\pi$
$619$	−6.73371	−	11.6631i	−0.270651	−	0.468781i	−0.969029	+	0.246948i	$0.920572\pi$
$620$		0			0
$621$	5.55673	+	8.73480i	0.222984	+	0.350516i
$622$		0			0
$623$	−4.53733	−	7.85889i	−0.181784	−	0.314860i
$624$		0			0
$625$	−9.79558	+	16.9664i	−0.391823	+	0.678657i
$626$		0			0
$627$	−6.36378	+	2.42004i	−0.254145	+	0.0966470i
$628$		0			0
$629$	−29.6485			−1.18216
$630$		0			0
$631$	43.2599			1.72215			0.861074	−	0.508480i	$-0.169792\pi$
$631$	43.2599			1.72215			0.861074	+	0.508480i	$0.169792\pi$
$632$		0			0
$633$	23.1328	+	18.8525i	0.919445	+	0.749321i
$634$		0			0
$635$	6.51202	−	11.2792i	0.258422	−	0.447600i
$636$		0			0
$637$	14.5325	+	25.1710i	0.575798	+	0.997311i
$638$		0			0
$639$	24.3407	+	8.07728i	0.962902	+	0.319532i
$640$		0			0
$641$	7.26741	+	12.5875i	0.287046	+	0.497178i	0.973103	−	0.230370i	$-0.0739935\pi$
$641$	7.26741	+	12.5875i	0.287046	+	0.497178i	−0.686058	+	0.727547i	$0.740660\pi$
$642$		0			0
$643$	−18.2849	+	31.6703i	−0.721085	+	1.24896i	0.239480	+	0.970901i	$0.423023\pi$
$643$	−18.2849	+	31.6703i	−0.721085	+	1.24896i	−0.960565	+	0.278055i	$0.910310\pi$
$644$		0			0
$645$	1.86805	−	11.5605i	0.0735543	−	0.455193i
$646$		0			0
$647$	17.5802			0.691149			0.345575	−	0.938391i	$-0.387684\pi$
$647$	17.5802			0.691149			0.345575	+	0.938391i	$0.387684\pi$
$648$		0			0
$649$	5.15369			0.202300
$650$		0			0
$651$	−1.32164	+	8.17903i	−0.0517992	+	0.320561i
$652$		0			0
$653$	−1.99448	+	3.45454i	−0.0780499	+	0.135186i	−0.902409	−	0.430882i	$-0.858203\pi$
$653$	−1.99448	+	3.45454i	−0.0780499	+	0.135186i	0.824359	+	0.566068i	$0.191536\pi$
$654$		0			0
$655$	−2.21631	−	3.83876i	−0.0865984	−	0.149993i
$656$		0			0
$657$	2.95257	+	0.979789i	0.115191	+	0.0382252i
$658$		0			0
$659$	−10.9405	−	18.9494i	−0.426179	−	0.738165i	0.570350	−	0.821402i	$-0.306807\pi$
$659$	−10.9405	−	18.9494i	−0.426179	−	0.738165i	−0.996530	+	0.0832371i	$0.973474\pi$
$660$		0			0
$661$	12.2552	−	21.2266i	0.476672	−	0.825620i	−0.522971	−	0.852351i	$-0.675176\pi$
$661$	12.2552	−	21.2266i	0.476672	−	0.825620i	0.999643	+	0.0267304i	$0.00850957\pi$
$662$		0			0
$663$	40.3965	+	32.9220i	1.56887	+	1.27858i
$664$		0			0
$665$	−0.584985			−0.0226848
$666$		0			0
$667$	−8.17981			−0.316724
$668$		0			0
$669$	13.6492	−	5.19055i	0.527707	−	0.200678i
$670$		0			0
$671$	−8.24532	+	14.2813i	−0.318307	+	0.551324i
$672$		0			0
$673$	9.67721	+	16.7614i	0.373029	+	0.646105i	0.990030	−	0.140857i	$-0.0449858\pi$
$673$	9.67721	+	16.7614i	0.373029	+	0.646105i	−0.617001	+	0.786962i	$0.711653\pi$
$674$		0			0
$675$	24.0375	−	1.03390i	0.925204	−	0.0397950i
$676$		0			0
$677$	−2.68613	−	4.65251i	−0.103236	−	0.178811i	0.809780	−	0.586734i	$-0.199586\pi$
$677$	−2.68613	−	4.65251i	−0.103236	−	0.178811i	−0.913016	+	0.407923i	$0.866253\pi$
$678$		0			0
$679$	5.13804	−	8.89935i	0.197180	−	0.341526i
$680$		0			0
$681$	−24.6360	+	9.36865i	−0.944053	+	0.359007i
$682$		0			0
$683$	−9.64173			−0.368931			−0.184465	−	0.982839i	$-0.559055\pi$
$683$	−9.64173			−0.368931			−0.184465	+	0.982839i	$0.559055\pi$
$684$		0			0
$685$	−10.8292			−0.413764
$686$		0			0
$687$	−29.2286	−	23.8204i	−1.11514	−	0.908806i
$688$		0			0
$689$	32.1953	−	55.7639i	1.22654	−	2.12444i
$690$		0			0
$691$	−3.70741	−	6.42142i	−0.141036	−	0.244282i	0.786851	−	0.617143i	$-0.211710\pi$
$691$	−3.70741	−	6.42142i	−0.141036	−	0.244282i	−0.927887	+	0.372861i	$0.878377\pi$
$692$		0			0
$693$	−8.47865	+	7.53880i	−0.322077	+	0.286375i
$694$		0			0
$695$	0.182191	+	0.315564i	0.00691090	+	0.0119700i
$696$		0			0
$697$	36.7733	−	63.6932i	1.39289	−	2.41255i
$698$		0			0
$699$	−1.74967	+	10.8279i	−0.0661785	+	0.409548i
$700$		0			0
$701$	38.8731			1.46822			0.734108	−	0.679033i	$-0.237601\pi$
$701$	38.8731			1.46822			0.734108	+	0.679033i	$0.237601\pi$
$702$		0			0
$703$	4.71508			0.177833
$704$		0			0
$705$	0.230932	−	1.42913i	0.00869739	−	0.0538241i
$706$		0			0
$707$	1.32349	−	2.29235i	0.0497749	−	0.0862127i
$708$		0			0
$709$	−7.93086	−	13.7367i	−0.297850	−	0.515891i	0.677794	−	0.735252i	$-0.262936\pi$
$709$	−7.93086	−	13.7367i	−0.297850	−	0.515891i	−0.975644	+	0.219361i	$0.929603\pi$
$710$		0			0
$711$	−7.57171	−	36.7500i	−0.283961	−	1.37823i
$712$		0			0
$713$	−4.95281	−	8.57852i	−0.185484	−	0.321268i
$714$		0			0
$715$	−5.71806	+	9.90397i	−0.213843	+	0.370387i
$716$		0			0
$717$	31.9263	+	26.0190i	1.19231	+	0.971698i
$718$		0			0
$719$	−38.4764			−1.43493			−0.717463	−	0.696596i	$-0.754697\pi$
$719$	−38.4764			−1.43493			−0.717463	+	0.696596i	$0.754697\pi$
$720$		0			0
$721$	−16.2894			−0.606648
$722$		0			0
$723$	−2.22355	+	0.845580i	−0.0826949	+	0.0314475i
$724$		0			0
$725$	−9.50516	+	16.4634i	−0.353013	+	0.611436i
$726$		0			0
$727$	19.3981	+	33.5985i	0.719435	+	1.24610i	0.961224	+	0.275769i	$0.0889325\pi$
$727$	19.3981	+	33.5985i	0.719435	+	1.24610i	−0.241789	+	0.970329i	$0.577734\pi$
$728$		0			0
$729$	−15.4579	−	22.1371i	−0.572515	−	0.819894i
$730$		0			0
$731$	34.9599	+	60.5524i	1.29304	+	2.23961i
$732$		0			0
$733$	−9.51171	+	16.4748i	−0.351323	+	0.608509i	−0.986482	−	0.163873i	$-0.947601\pi$
$733$	−9.51171	+	16.4748i	−0.351323	+	0.608509i	0.635159	+	0.772382i	$0.280935\pi$
$734$		0			0
$735$	5.97940	−	2.27387i	0.220554	−	0.0838728i
$736$		0			0
$737$	47.2327			1.73984
$738$		0			0
$739$	10.4137			0.383073			0.191536	−	0.981485i	$-0.438653\pi$
$739$	10.4137			0.383073			0.191536	+	0.981485i	$0.438653\pi$
$740$		0			0
$741$	−6.42435	−	5.23566i	−0.236004	−	0.192337i
$742$		0			0
$743$	−11.4220	+	19.7835i	−0.419033	+	0.725786i	−0.995842	−	0.0910923i	$-0.970964\pi$
$743$	−11.4220	+	19.7835i	−0.419033	+	0.725786i	0.576809	+	0.816879i	$0.304297\pi$
$744$		0			0
$745$	5.93129	+	10.2733i	0.217306	+	0.376384i
$746$		0			0
$747$	−6.99012	−	33.9271i	−0.255755	−	1.24133i
$748$		0			0
$749$	−4.91935	−	8.52057i	−0.179749	−	0.311335i
$750$		0			0
$751$	20.5835	−	35.6516i	0.751101	−	1.30095i	−0.196188	−	0.980566i	$-0.562856\pi$
$751$	20.5835	−	35.6516i	0.751101	−	1.30095i	0.947289	−	0.320379i	$-0.103810\pi$
$752$		0			0
$753$	2.18870	−	13.5448i	0.0797606	−	0.493601i
$754$		0			0
$755$	8.33021			0.303167
$756$		0			0
$757$	9.33585			0.339317			0.169659	−	0.985503i	$-0.445733\pi$
$757$	9.33585			0.339317			0.169659	+	0.985503i	$0.445733\pi$
$758$		0			0
$759$	2.16383	−	13.3909i	0.0785420	−	0.486060i
$760$		0			0
$761$	19.1184	−	33.1141i	0.693042	−	1.20038i	−0.277794	−	0.960641i	$-0.589603\pi$
$761$	19.1184	−	33.1141i	0.693042	−	1.20038i	0.970836	−	0.239744i	$-0.0770634\pi$
$762$		0			0
$763$	−0.388199	−	0.672380i	−0.0140537	−	0.0243418i
$764$		0			0
$765$	8.57163	−	7.62147i	0.309908	−	0.275555i
$766$		0			0
$767$	3.13670	+	5.43292i	0.113260	+	0.196171i
$768$		0			0
$769$	−7.96672	+	13.7988i	−0.287287	+	0.497596i	−0.973161	−	0.230124i	$-0.926087\pi$
$769$	−7.96672	+	13.7988i	−0.287287	+	0.497596i	0.685874	+	0.727720i	$0.259420\pi$
$770$		0			0
$771$	24.3670	+	19.8584i	0.877556	+	0.715183i
$772$		0			0
$773$	31.4709			1.13193			0.565964	−	0.824430i	$-0.308504\pi$
$773$	31.4709			1.13193			0.565964	+	0.824430i	$0.308504\pi$
$774$		0			0
$775$	−23.0212			−0.826946
$776$		0			0
$777$	7.34411	−	2.79284i	0.263468	−	0.100193i
$778$		0			0
$779$	−5.84815	+	10.1293i	−0.209532	+	0.362919i
$780$		0			0
$781$	−16.8016	−	29.1012i	−0.601208	−	1.04132i
$782$		0			0
$783$	21.3138	−	0.916752i	0.761693	−	0.0327620i
$784$		0			0
$785$	−0.282827	−	0.489870i	−0.0100945	−	0.0174842i
$786$		0			0
$787$	5.19905	−	9.00502i	0.185326	−	0.320994i	−0.758360	−	0.651836i	$-0.773999\pi$
$787$	5.19905	−	9.00502i	0.185326	−	0.320994i	0.943686	+	0.330841i	$0.107333\pi$
$788$		0			0
$789$	−31.0363	+	11.8026i	−1.10492	+	0.420182i
$790$		0			0
$791$	−2.45267			−0.0872070
$792$		0			0
$793$	−20.0734			−0.712829
$794$		0			0
$795$	−10.9860	−	8.95330i	−0.389634	−	0.317541i
$796$		0			0
$797$	−21.5596	+	37.3423i	−0.763680	+	1.32273i	0.177262	+	0.984164i	$0.443276\pi$
$797$	−21.5596	+	37.3423i	−0.763680	+	1.32273i	−0.940942	+	0.338569i	$0.890057\pi$
$798$		0			0
$799$	4.32182	+	7.48561i	0.152895	+	0.264822i
$800$		0			0
$801$	−26.8564	−	8.91211i	−0.948924	−	0.314894i
$802$		0			0
$803$	−2.03806	−	3.53003i	−0.0719217	−	0.124572i
$804$		0			0
$805$	0.582744	−	1.00934i	0.0205390	−	0.0355747i
$806$		0			0
$807$	2.02878	−	12.5552i	0.0714164	−	0.441963i
$808$		0			0
$809$	−50.1645			−1.76369			−0.881844	−	0.471541i	$-0.843698\pi$
$809$	−50.1645			−1.76369			−0.881844	+	0.471541i	$0.843698\pi$
$810$		0			0
$811$	6.96688			0.244640			0.122320	−	0.992491i	$-0.460967\pi$
$811$	6.96688			0.244640			0.122320	+	0.992491i	$0.460967\pi$
$812$		0			0
$813$	1.62569	−	10.0606i	0.0570155	−	0.352842i
$814$		0			0
$815$	3.33048	−	5.76856i	0.116662	−	0.202064i
$816$		0			0
$817$	−5.55976	−	9.62979i	−0.194512	−	0.336904i
$818$		0			0
$819$	−13.1076	−	4.34968i	−0.458018	−	0.151990i
$820$		0			0
$821$	−14.7056	−	25.4708i	−0.513228	−	0.888937i	−0.999882	−	0.0153424i	$-0.995116\pi$
$821$	−14.7056	−	25.4708i	−0.513228	−	0.888937i	0.486654	−	0.873595i	$-0.338217\pi$
$822$		0			0
$823$	−9.26445	+	16.0465i	−0.322938	+	0.559346i	−0.981093	−	0.193537i	$-0.938004\pi$
$823$	−9.26445	+	16.0465i	−0.322938	+	0.559346i	0.658154	+	0.752883i	$0.271337\pi$
$824$		0			0
$825$	−24.4373	−	19.9157i	−0.850799	−	0.693377i
$826$		0			0
$827$	16.7219			0.581477			0.290738	−	0.956803i	$-0.406099\pi$
$827$	16.7219			0.581477			0.290738	+	0.956803i	$0.406099\pi$
$828$		0			0
$829$	33.8306			1.17499			0.587493	−	0.809229i	$-0.300115\pi$
$829$	33.8306			1.17499			0.587493	+	0.809229i	$0.300115\pi$
$830$		0			0
$831$	−4.74863	+	1.80582i	−0.164728	+	0.0626434i
$832$		0			0
$833$	−19.0979	+	33.0785i	−0.661703	+	1.14610i
$834$		0			0
$835$	−0.418562	−	0.724970i	−0.0144849	−	0.0250886i
$836$		0			0
$837$	13.8668	+	21.7976i	0.479306	+	0.753436i
$838$		0			0
$839$	−10.4325	−	18.0695i	−0.360168	−	0.623830i	0.627820	−	0.778359i	$-0.283947\pi$
$839$	−10.4325	−	18.0695i	−0.360168	−	0.623830i	−0.987988	+	0.154529i	$0.950614\pi$
$840$		0			0
$841$	6.07188	−	10.5168i	0.209375	−	0.362648i
$842$		0			0
$843$	−22.9733	+	8.73635i	−0.791242	+	0.300896i
$844$		0			0
$845$	−6.01636			−0.206969
$846$		0			0
$847$	4.28271			0.147156
$848$		0			0
$849$	5.10057	+	4.15681i	0.175051	+	0.142661i
$850$		0			0
$851$	−4.69702	+	8.13547i	−0.161012	+	0.278880i
$852$		0			0
$853$	−9.63115	−	16.6816i	−0.329764	−	0.571168i	0.652701	−	0.757616i	$-0.273636\pi$
$853$	−9.63115	−	16.6816i	−0.329764	−	0.571168i	−0.982465	+	0.186448i	$0.940303\pi$
$854$		0			0
$855$	−1.36317	+	1.21206i	−0.0466193	+	0.0414516i
$856$		0			0
$857$	−4.48193	−	7.76293i	−0.153100	−	0.265177i	0.779266	−	0.626694i	$-0.215592\pi$
$857$	−4.48193	−	7.76293i	−0.153100	−	0.265177i	−0.932365	+	0.361517i	$0.882259\pi$
$858$		0			0
$859$	4.59070	−	7.95133i	0.156633	−	0.271296i	−0.777020	−	0.629476i	$-0.783269\pi$
$859$	4.59070	−	7.95133i	0.156633	−	0.271296i	0.933652	+	0.358180i	$0.116603\pi$
$860$		0			0
$861$	−3.10916	+	19.2411i	−0.105960	+	0.655736i
$862$		0			0
$863$	−29.6856			−1.01051			−0.505255	−	0.862970i	$-0.668602\pi$
$863$	−29.6856			−1.01051			−0.505255	+	0.862970i	$0.668602\pi$
$864$		0			0
$865$	8.30288			0.282307
$866$		0			0
$867$	−6.22752	+	38.5392i	−0.211498	+	1.30886i
$868$		0			0
$869$	−24.5820	+	42.5773i	−0.833888	+	1.44434i
$870$		0			0
$871$	28.7473	+	49.7918i	0.974065	+	1.68713i
$872$		0			0
$873$	−6.46605	−	31.3835i	−0.218843	−	1.06217i
$874$		0			0
$875$	−2.81679	−	4.87883i	−0.0952250	−	0.164934i
$876$		0			0
$877$	−2.69804	+	4.67314i	−0.0911064	+	0.157801i	−0.907977	−	0.419020i	$-0.862374\pi$
$877$	−2.69804	+	4.67314i	−0.0911064	+	0.157801i	0.816871	+	0.576821i	$0.195707\pi$
$878$		0			0
$879$	13.8287	+	11.2700i	0.466431	+	0.380128i
$880$		0			0
$881$	15.9924			0.538798			0.269399	−	0.963029i	$-0.413175\pi$
$881$	15.9924			0.538798			0.269399	+	0.963029i	$0.413175\pi$
$882$		0			0
$883$	−26.0889			−0.877961			−0.438981	−	0.898497i	$-0.644660\pi$
$883$	−26.0889			−0.877961			−0.438981	+	0.898497i	$0.644660\pi$
$884$		0			0
$885$	1.29060	−	0.490792i	0.0433829	−	0.0164978i
$886$		0			0
$887$	4.88522	−	8.46145i	0.164030	−	0.284108i	−0.772281	−	0.635281i	$-0.780884\pi$
$887$	4.88522	−	8.46145i	0.164030	−	0.284108i	0.936310	+	0.351174i	$0.114217\pi$
$888$		0			0
$889$	−10.3041	−	17.8472i	−0.345588	−	0.598576i
$890$		0			0
$891$	−4.13741	+	35.1347i	−0.138609	+	1.17706i
$892$		0			0
$893$	−0.687309	−	1.19045i	−0.0229999	−	0.0398370i
$894$		0			0
$895$	−2.74829	+	4.76018i	−0.0918653	+	0.159115i
$896$		0			0
$897$	15.4334	−	5.86907i	0.515307	−	0.195963i
$898$		0			0
$899$	−20.4126			−0.680800
$900$		0			0
$901$	84.6191			2.81907
$902$		0			0
$903$	−14.3637	−	11.7060i	−0.477994	−	0.389551i
$904$		0			0
$905$	−0.734186	+	1.27165i	−0.0244052	+	0.0422710i
$906$		0			0
$907$	−3.09761	−	5.36522i	−0.102854	−	0.178149i	0.810005	−	0.586423i	$-0.199464\pi$
$907$	−3.09761	−	5.36522i	−0.102854	−	0.178149i	−0.912860	+	0.408274i	$0.866131\pi$
$908$		0			0
$909$	−1.66557	−	8.08397i	−0.0552434	−	0.268129i
$910$		0			0
$911$	−4.06086	−	7.03362i	−0.134542	−	0.233034i	0.790880	−	0.611971i	$-0.209623\pi$
$911$	−4.06086	−	7.03362i	−0.134542	−	0.233034i	−0.925423	+	0.378937i	$0.876290\pi$
$912$		0			0
$913$	−22.6938	+	39.3069i	−0.751056	+	1.30087i
$914$		0			0
$915$	−0.704782	+	4.36157i	−0.0232994	+	0.144189i
$916$		0			0
$917$	−7.01381			−0.231616
$918$		0			0
$919$	17.9176			0.591046			0.295523	−	0.955336i	$-0.404506\pi$
$919$	17.9176			0.591046			0.295523	+	0.955336i	$0.404506\pi$
$920$		0			0
$921$	2.60507	−	16.1216i	0.0858401	−	0.531224i
$922$		0			0
$923$	20.4520	−	35.4238i	0.673184	−	1.16599i
$924$		0			0
$925$	10.9161	+	18.9073i	0.358920	+	0.621667i
$926$		0			0
$927$	−37.9584	+	33.7508i	−1.24672	+	1.10852i
$928$		0			0
$929$	−18.7818	−	32.5310i	−0.616210	−	1.06731i	−0.990171	−	0.139863i	$-0.955334\pi$
$929$	−18.7818	−	32.5310i	−0.616210	−	1.06731i	0.373961	−	0.927445i	$-0.377999\pi$
$930$		0			0
$931$	3.03718	−	5.26056i	0.0995397	−	0.172408i
$932$		0			0
$933$	−27.2834	−	22.2352i	−0.893219	−	0.727948i
$934$		0			0
$935$	−15.0288			−0.491494
$936$		0			0
$937$	8.17412			0.267037			0.133518	−	0.991046i	$-0.457372\pi$
$937$	8.17412			0.267037			0.133518	+	0.991046i	$0.457372\pi$
$938$		0			0
$939$	−24.3644	+	9.26536i	−0.795101	+	0.302363i
$940$		0			0
$941$	−30.0404	+	52.0315i	−0.979288	+	1.69618i	−0.314299	+	0.949324i	$0.601769\pi$
$941$	−30.0404	+	52.0315i	−0.979288	+	1.69618i	−0.664989	+	0.746853i	$0.731564\pi$
$942$		0			0
$943$	−11.6515	−	20.1810i	−0.379425	−	0.657183i
$944$		0			0
$945$	−1.40531	+	2.69531i	−0.0457148	+	0.0876786i
$946$		0			0
$947$	−2.42105	−	4.19339i	−0.0786737	−	0.136267i	0.824004	−	0.566584i	$-0.191735\pi$
$947$	−2.42105	−	4.19339i	−0.0786737	−	0.136267i	−0.902678	+	0.430317i	$0.858402\pi$
$948$		0			0
$949$	2.48086	−	4.29697i	0.0805320	−	0.139486i
$950$		0			0
$951$	15.6447	−	5.94943i	0.507315	−	0.192923i
$952$		0			0
$953$	0.318996			0.0103333			0.00516665	−	0.999987i	$-0.498355\pi$
$953$	0.318996			0.0103333			0.00516665	+	0.999987i	$0.498355\pi$
$954$		0			0
$955$	1.98176			0.0641282
$956$		0			0
$957$	−21.6683	−	17.6591i	−0.700438	−	0.570836i
$958$		0			0
$959$	−8.56764	+	14.8396i	−0.276663	+	0.479195i
$960$		0			0
$961$	3.14030	+	5.43917i	0.101300	+	0.175457i
$962$		0			0
$963$	−29.1176	−	9.66247i	−0.938301	−	0.311369i
$964$		0			0
$965$	−2.23506	−	3.87123i	−0.0719490	−	0.124619i
$966$		0			0
$967$	−2.22781	+	3.85868i	−0.0716415	+	0.124087i	−0.899621	−	0.436672i	$-0.856157\pi$
$967$	−2.22781	+	3.85868i	−0.0716415	+	0.124087i	0.827979	+	0.560759i	$0.189490\pi$
$968$		0			0
$969$	1.73736	−	10.7517i	0.0558121	−	0.345395i
$970$		0			0
$971$	0.821485			0.0263627			0.0131814	−	0.999913i	$-0.495804\pi$
$971$	0.821485			0.0263627			0.0131814	+	0.999913i	$0.495804\pi$
$972$		0			0
$973$	0.576568			0.0184839
$974$		0			0
$975$	6.12143	−	37.8827i	0.196043	−	1.21322i
$976$		0			0
$977$	−25.9672	+	44.9765i	−0.830764	+	1.43893i	0.0666694	+	0.997775i	$0.478763\pi$
$977$	−25.9672	+	44.9765i	−0.830764	+	1.43893i	−0.897433	+	0.441150i	$0.854571\pi$
$978$		0			0
$979$	18.5381	+	32.1090i	0.592481	+	1.02621i
$980$		0			0
$981$	−2.29774	−	0.762489i	−0.0733612	−	0.0243444i
$982$		0			0
$983$	−3.61076	−	6.25402i	−0.115165	−	0.199472i	0.802680	−	0.596409i	$-0.203406\pi$
$983$	−3.61076	−	6.25402i	−0.115165	−	0.199472i	−0.917846	+	0.396937i	$0.870073\pi$
$984$		0			0
$985$	−3.53270	+	6.11882i	−0.112561	+	0.194962i
$986$		0			0
$987$	−1.77567	−	1.44712i	−0.0565202	−	0.0460623i
$988$		0			0
$989$	22.1539			0.704452
$990$		0			0
$991$	−26.4072			−0.838853			−0.419426	−	0.907789i	$-0.637769\pi$
$991$	−26.4072			−0.838853			−0.419426	+	0.907789i	$0.637769\pi$
$992$		0			0
$993$	−3.64909	+	1.38769i	−0.115800	+	0.0440369i
$994$		0			0
$995$	−7.68665	+	13.3137i	−0.243683	+	0.422072i
$996$		0			0
$997$	9.37147	+	16.2319i	0.296797	+	0.514068i	0.975401	−	0.220436i	$-0.0707481\pi$
$997$	9.37147	+	16.2319i	0.296797	+	0.514068i	−0.678604	+	0.734504i	$0.737415\pi$
$998$		0			0
$999$	11.3270	−	21.7247i	0.358372	−	0.687338i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	684.2.i.d.229.5	✓	16
3.2	odd	2		2052.2.i.c.685.5		16
9.2	odd	6		2052.2.i.c.1369.5		16
9.4	even	3		6156.2.a.o.1.5		8
9.5	odd	6		6156.2.a.t.1.4		8
9.7	even	3	inner	684.2.i.d.457.5	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
684.2.i.d.229.5	✓	16	1.1	even	1	trivial
684.2.i.d.457.5	yes	16	9.7	even	3	inner
2052.2.i.c.685.5		16	3.2	odd	2
2052.2.i.c.1369.5		16	9.2	odd	6
6156.2.a.o.1.5		8	9.4	even	3
6156.2.a.t.1.4		8	9.5	odd	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	684.i (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.276297 + 1.70987i) q^{3} +(0.304016 - 0.526570i) q^{5} +(-0.481049 - 0.833201i) q^{7} +(-2.84732 - 0.944864i) q^{9} +O(q^{10})\)	\(q+(-0.276297 + 1.70987i) q^{3} +(0.304016 - 0.526570i) q^{5} +(-0.481049 - 0.833201i) q^{7} +(-2.84732 - 0.944864i) q^{9} +(1.96542 + 3.40420i) q^{11} +(-2.39243 + 4.14380i) q^{13} +(0.816369 + 0.665317i) q^{15} -6.28802 q^{17} +1.00000 q^{19} +(1.55758 - 0.592321i) q^{21} +(-0.996169 + 1.72542i) q^{23} +(2.31515 + 4.00996i) q^{25} +(2.40230 - 4.60749i) q^{27} +(2.05282 + 3.55558i) q^{29} +(-2.48593 + 4.30576i) q^{31} +(-6.36378 + 2.42004i) q^{33} -0.584985 q^{35} +4.71508 q^{37} +(-6.42435 - 5.23566i) q^{39} +(-5.84815 + 10.1293i) q^{41} +(-5.55976 - 9.62979i) q^{43} +(-1.36317 + 1.21206i) q^{45} +(-0.687309 - 1.19045i) q^{47} +(3.03718 - 5.26056i) q^{49} +(1.73736 - 10.7517i) q^{51} -13.4572 q^{53} +2.39007 q^{55} +(-0.276297 + 1.70987i) q^{57} +(0.655547 - 1.13544i) q^{59} +(2.09760 + 3.63315i) q^{61} +(0.582439 + 2.82692i) q^{63} +(1.45467 + 2.51956i) q^{65} +(6.00798 - 10.4061i) q^{67} +(-2.67500 - 2.18005i) q^{69} -8.54862 q^{71} -1.03696 q^{73} +(-7.49618 + 2.85067i) q^{75} +(1.89092 - 3.27517i) q^{77} +(6.25364 + 10.8316i) q^{79} +(7.21447 + 5.38066i) q^{81} +(5.77329 + 9.99963i) q^{83} +(-1.91166 + 3.31109i) q^{85} +(-6.64678 + 2.52766i) q^{87} +9.43217 q^{89} +4.60350 q^{91} +(-6.67544 - 5.44029i) q^{93} +(0.304016 - 0.526570i) q^{95} +(5.34045 + 9.24994i) q^{97} +(-2.37966 - 11.5499i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - q^{3} + 6 q^{5} - 5 q^{7} + 15 q^{9}+O(q^{10}) \) 16 * q - q^3 + 6 * q^5 - 5 * q^7 + 15 * q^9	\( 16 q - q^{3} + 6 q^{5} - 5 q^{7} + 15 q^{9} + 13 q^{11} - 3 q^{13} + 6 q^{15} - 20 q^{17} + 16 q^{19} + 3 q^{21} + 16 q^{23} - 14 q^{25} + 20 q^{27} + 3 q^{29} - 6 q^{31} - 17 q^{33} + 16 q^{37} - 16 q^{39} + 7 q^{41} - 3 q^{43} + 21 q^{45} + 10 q^{47} + 3 q^{49} - 2 q^{51} + 2 q^{53} - 12 q^{55} - q^{57} + 28 q^{59} - 7 q^{61} + 49 q^{63} + 21 q^{65} - 6 q^{67} - 56 q^{69} - 40 q^{71} - 20 q^{73} - 7 q^{75} + 22 q^{77} + 7 q^{79} + 39 q^{81} + 17 q^{83} + 24 q^{85} - 27 q^{87} - 32 q^{89} - 24 q^{91} - 22 q^{93} + 6 q^{95} + 11 q^{97} - q^{99}+O(q^{100}) \) 16 * q - q^3 + 6 * q^5 - 5 * q^7 + 15 * q^9 + 13 * q^11 - 3 * q^13 + 6 * q^15 - 20 * q^17 + 16 * q^19 + 3 * q^21 + 16 * q^23 - 14 * q^25 + 20 * q^27 + 3 * q^29 - 6 * q^31 - 17 * q^33 + 16 * q^37 - 16 * q^39 + 7 * q^41 - 3 * q^43 + 21 * q^45 + 10 * q^47 + 3 * q^49 - 2 * q^51 + 2 * q^53 - 12 * q^55 - q^57 + 28 * q^59 - 7 * q^61 + 49 * q^63 + 21 * q^65 - 6 * q^67 - 56 * q^69 - 40 * q^71 - 20 * q^73 - 7 * q^75 + 22 * q^77 + 7 * q^79 + 39 * q^81 + 17 * q^83 + 24 * q^85 - 27 * q^87 - 32 * q^89 - 24 * q^91 - 22 * q^93 + 6 * q^95 + 11 * q^97 - q^99

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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